 In this video, consider the 2 by 5 matrix A illustrated as shown, and for convenience, the RREF of this matrix is also included. One might normally have to compute it, but let's just pretend we threw in a calculator and it did it for us. Great. So given this matrix A, we want to find a basis for the column space and the null space of A, and I actually want to do it relatively quickly. To find the column space of A, you have to identify the pivot columns of the matrix, which you'll need an echelon form to do that, RREF will do that. The number of pivots is going to give you the rank of the matrix, that is, this is going to be the number of vectors you need to span the column space. So the column space, which is the span of all five columns, you only need two of the columns, and which two columns are you going to choose? You're going to choose the first column because that was a dependent variable. You get a pivot there in the matrix. So we're going to get 1 and negative 2, and then you're also going to take the fourth column because that was a pivot column, and so that gives us the vector 2 and 5. And so that's all that one has to do to find a basis for the column space of the matrix. If you have the RREF, you can do it in less than three seconds, basically. It's the row reduction that's really the expensive part there. Well, what if we want to find the null space? That is, we want to find a basis for the null space of the matrix here. So the null space will be the span of some vectors, but what vectors can we choose here? Well, one thing to mention is that the null space, we're going to compute its basis by looking at the non-pivot columns of the matrix. Counting those non-pivots, we see that the nullity of this matrix is going to be three. The nullity is counting how many free variables are there in the system. Each free variable will give us a spanner for the null space. And so let's now set up our template. We're going to get three vectors for the basis of the null space, like so. We get a vector for each of the free variables. So we have the free variable x2, x3. I can't really read that very well. x2, x3, and x5. So what we're going to do is we're going to put a star whenever there's a dependent variable. We're going to put a one in the spot indexed by the variable we're considering. Let's do like x2 first. And then we're going to put a zero for the other free variables. So the first position gets a star. The second position gets one because we're doing x2 right now. Then x3 is a free variable, put a zero. x4 is a dependent variable, put a star. And then x5 is a free variable, put a zero. Now the second vector, we're going to do this for x3. So we get a star for the dependent variable, zero for x2, one for x3, star for x4, and then zero for x5. And then the last vector is going to correspond to the free variable x5. We get a star for x1, zero for x2, zero for x3, star for x4, and one for x5. Now we have to come in and fill in the stars. Let's look at the first row. So we are going to look at the first row of the matrix. You're going to look at the numbers and switch the signs. So in the column for x2, we see a negative 3, so we record positive 3. In the column for x3, we see a 4, so we record a negative 4. In the column for x5, we see a negative 1 ninth, so we replace it with a positive 1 ninth. And we're going to do the same thing for the fourth row, focusing on that one right here. When you look in the column for x2, you see a zero, so I'm going to put a negative zero, which is still zero. Same thing for x3, you see a zero, so you get a zero there. We have to switch the sign, but zero doesn't have a sign, so switching it doesn't do anything. When you look at the fifth column, you see a negative 1 ninth, so you're going to record 1 ninth right there. And now we have a basis for the null space. 3, 1, 0, 0, 0, negative 4, 0, 1, 0, 0, and 1 ninth, 0, 0, 1 ninth, 1. Now, one thing I do want to mention is that if you don't like having fractions, some of us suffer from a disease called ratiophobia. The disease, you know, this is the fear of fractions, right? If you don't like fractions, that's okay. We can get rid of them because when you look at the vector 1 ninth, 0, 0, 1 ninth, and 1. Notice we could factor out the scalar 1 ninth, that's just a real number. And that gives us 1, 0, 0, 1, and 9. I want you to be aware that if you were to take a vector like this one and you swap it out for some scalar multiple, linear independence is not affected by that. If you replace a vector with a scalar multiple of each other, if that's that was independent, it would stay independent. If it was dependent, it would stay dependent. Swapping out a stretch or dilation or compression of a vector will not change independence. That's also true for spanning. If I have a vector that points in one direction two meters and I shrink it to be one meter, I still have that direction. That's what matters, the direction you're pointing in. And so as such, you can actually replace, you can replace this vector with the following. Basically, you can just scale it by 9, in which case you're going to get 1, 0, 0, 1, and 9. And that's an alternative basis you could use. It has the advantage that it doesn't have fractions anymore, which fractions we know can be a little bit troublesome when it comes to arithmetic. So I mentioned that as an example, because bases are not unique. A base is just an independent spanning set. Every subspace has multiple spanning sets, and therefore any of those different spanning sets can be pruned down into different independent bases. So therefore, there's some options, right? So if you want the option to not have a fraction, you can actually choose that like we did in this example. And so I showed you in this video how to find a basis for the column space and the null space really quickly. Like the whole video is only about six minutes right now. It took us about 30 seconds for the first one. And really, we could do the null space in less than a minute as well if we had the RREF, which calculators can help us in that regard. And so that brings us to the end of section 2.7 about bases. We focused, although we mentioned general bases, we focused a lot on how to compute a basis for the column space and null space. Because it turns out if we can find a basis for the column space and a null space, we can basically find the basis for any subspace in existence. Because every vector space, every subspace can be realized as the column space of some matrix or the null space of some matrix. As long as we can put things into coordinates, what's a coordinate? I think we'll talk about that in the next lecture. So please, when you're ready, turn that one on and we can chat some more about coordinate vectors. See you, everyone. Bye.